1. Field
The present application relates to map projections of the world or any land or other area. Specifically, it relates to improved maps in the Azimuthal Equidistant projection format.
2. Prior Art
Although there are hundreds of available map projections, the majority of world maps in atlases and books are still drawn by the Mercator projection or its variations. In such maps, the equator is placed in the center and the North Pole at the top. As a result of being raised with such equator-centered maps, we have a tendency to think horizontally or along latitude lines. Thus,                We may think Hawaii is directly between the U.S. and Japan.        If we go to India, China, or any other Asian country by an eastbound flight from the East Coast of the U.S., we assume that we would fly over the Mediterranean Sea and the Middle Eastern countries.        But if we depart from the West-Coast cities to those destinations by a westbound flight, we assume that we would fly westward, crossing the Pacific Ocean.        We assume that the distance between Seattle and Moscow is greater than that between Miami and Moscow.        We assume that if we went due west from San Francisco we would cross over Japan and likewise if we went due east we would pass over somewhere near Washington D.C. In other words, we assume that east-west line of cardinal direction is parallel to latitude line.        
However, all of the above assumptions are wrong: the Aleutian Islands, not Hawaii, are between the U.S. and Japan. From most cities of the U.S. to India or China, we would cross the Arctic Ocean, not the Pacific or the Atlantic Ocean. The distance between Miami and Moscow is slightly longer than between Seattle and Moscow; this is because the shortest air route of the latter is again over the Arctic Ocean, not over the North American Continent. I will discuss the East-West paths in conjunction with FIG. 5 in the detailed description section.
David Greenhood, in “Down to Earth: Mapping for Everybody” (Holiday House, 1944, repub. by U. of Chicago Press 1964 as “Mapping”) notes similar misconceptions. Greenhood even suggests we discard all scientific world maps. However, more than 60 years later, misleading world maps still prevail.
World maps in Western civilization were developed mainly for mariners, who were not very concerned about shortening a day out of a month-long windjammer cruise by selecting the shortest possible course. Quite a time has elapsed since then and the age of air has mostly supplanted ocean travel. But peoples' mindsets remain the same: they cannot shed their adherence to the traditional equator-centered maps.
Nowadays in the dawning of the space age, a virtual Earth presented on a computer is popular. Composed from satellite pictures, its appearance is very realistic. By manipulating a mouse, it gives us a feeling of flying over the real Earth. We can draw arc lines to show air routes between many cities in the world. But functionally, it is nothing but a flat projection of a 3-D globe. Even the globe itself has problems: Although a globe is the only possible distortion-free representation of the Earth, we cannot see all parts of the whole world in one view. To view relationships between remote countries, we have to turn it around, back and forth. Even when viewing any area of the Earth, figures in the central portion look expanded and the ones in the outer or surrounding areas are shrunk or invisible.
Therefore, we need world maps more suited for the age of aviation. Among the existing world map projections, I believe the Azimuthal Equidistant (AE) projection is optimal. Azimuthal projections preserve azimuths (angles or directions) and roundness in general. Also the azimuthal equidistant projection has linearly-increasing distances from a central point of a map. In this projection a straight line from a central point to any other point gives the shortest distance between the two points. Using it, we can show the correct distance and direction from a center point on the map to any place in the world. Because of these features, this projection has been used for airline routes, seismic propagation, radio communication, and others. However, presently available maps drawn by this projection have a serious defect. The shapes of continents at the periphery are stretched like a figure in a mirror of a fun house. Since the shapes are too different from real figures on a globe to be recognized, this map is too grotesque for daily use.
Incidentally, although we can move a virtual Earth on a computer anyway we want, similar to the manipulation of a globe, I have not seen a flat world map that is movable and generally usable like a virtual Earth. Because of distortion problems that are contained in most map projections, a moving flat map is unpractical. If a center point of such a virtual flat map in the Mercator projection were moved gradually to the north or south, one would see the continents change their shapes and sizes. I.e., they would be dilated then shrunk constantly on computer. I will discuss this matter later in relation to a satellite path and the rotation of the Earth.
In the AE projection, an antipodal point (a point directly or exactly opposite a given point on Earth) is not shown as a point, but is transformed and stretched. It is actually shown as a line of the full circumference of a map. Thus the antipodal point is not apparent, not only in this projection but also in almost all projections. It may be not so important to know its location, but if we can show the antipodal point of our hometown clearly in maps, it will pique peoples' interests, especially among young students, toward geography.
In order to overcome these problems, my above-referenced patent provides a good solution. Since the purpose of the embodiments shown in this patent was to provide a unified shape of all continents in less distortion, the center point of the continental hemisphere could not be moved away from a certain point in France. In the resulting map, some countries, such as Japan, are located at the farthest border of a circle. This map may be less appealing as a reference world map to people who live in such countries at the periphery. Besides, it lacks an important aspect of the AE projection: Except for France and its neighboring countries, the shortest air routes between other countries are not shown by straight and measurable lines, but are only suggested by graticule lines of latitude and longitude.
In conclusion, there are many prior-art maps in geography and cartography but they are not satisfactory because                1. The majority of popular world maps has the equator at its center and has misled peoples' conceptions about the shortest air routes between most of the landmasses (I use this word for “continents and major islands” in this application).        2. An AE projection map shows the correct distance and direction between a given center point of the map and any other point on Earth, but deformation of shapes of landmasses increases and becomes unrecognizable toward the edges rim.        3. The AE projection and others do not show the antipodal point clearly.        